Directions: Use this worksheet to record your answers to the questions or

Question

problems for the Graphs of Polynomials portfolio. When you are finished, save this
worksheet with your answers and submit it for a portfolio grade.
Where indicated, draw figures on a blank sheet of paper or on a sheet of graph paper.
Be sure to label each sheet so that your teacher knows which answer goes with which
question. You can scan these figures and submit them as individual documents with
your portfolio worksheet, or you can take pictures of them and insert the pictures on
the portfolio worksheet. Finding the Zeros of a Polynomial Function
A. Choose whole numbers to fill in the two blanks in the following polynomial
function: h(x)=-3x^2(5x-------+)(x- -------)
C. Plot the zeros of h(x) on a coordinate plane on a piece of graph paper. Label
this paper Graph 1.
Note: You will complete Graph 1 throughout the portfolio. Do not worry about
scaling the y-axis at this time.

Answer 1

To fill in the blanks of the polynomial function \( h(x) = -3x^2(5x - ____)(x - ____) \), we need to choose whole numbers for the coefficients of the polynomial. Let's select the following numbers:

1. For the first blank, let's choose 2.
2. For the second blank, let's choose 3.

The completed polynomial function will be:

\[ h(x) = -3x^2(5x - 2)(x - 3) \]

Now, let's find the zeros of the polynomial function:

1. Set \( h(x) = 0 \): \[ -3x^2(5x - 2)(x - 3) = 0 \]

2. This equation will equal zero if any of the factors equal zero:

   • \( -3x^2 = 0 \) → \( x = 0 \)
   • \( 5x - 2 = 0 \) → \( 5x = 2 \) → \( x = \frac{2}{5} \)
   • \( x - 3 = 0 \) → \( x = 3 \)

So, the zeros of the function \( h(x) \) are:

1. \( x = 0 \)
2. \( x = \frac{2}{5} \)
3. \( x = 3 \)

C. Plotting the Zeros on a Coordinate Plane

Now, let's plot these zeros on a coordinate plane:

1. \( (0, 0) \) → The point at the origin.
2. \( \left(\frac{2}{5}, 0\right) \) → This is approximately at \( (0.4, 0) \).
3. \( (3, 0) \) → This point is located at \( (3, 0) \).

You can label these points as follows:

• Point A: \( (0, 0) \)
• Point B: \( \left(\frac{2}{5}, 0\right) \)
• Point C: \( (3, 0) \)

Make sure to provide appropriate labeling on your Graph 1, and you can indicate that the x-axis represents the values of \( x \) while the y-axis is simply the output of the polynomial at those values (which is 0 for these zeros).

Once your graph is created and labeled, either scan it or take a picture for submission along with this worksheet!